## Friday, June 3, 2016

### The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

Accurate digits are highlighted in green.  Calculations are used with a TI 84 Plus CE.

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4 + … = Σ(x^n/n!, from n = 0 to ∞)

 x = e^x 10 terms 25 terms 50 terms 1 2.718281828 2.718281801 2.718281828 2.718281828 3 20.08553692 20.07966518 20.08553692 20.08553692 5 148.4131591 146.380601 148.4131591 148.4131591 9.9 19930.37044 11869.50538 19930.07221 19930.37044

e^(-x) = 1 – x + x^2/2! – x^3/3! + x^4/4 - … = Σ( (-x)^n/n!, from n = 0 to ∞)

 x = e^(-x) 10 terms 25 terms 50 terms 1 0.3678794412 0.3678794643 0.3678794412 0.3678794412 3 0.0497870684 0.0533258929 0.0497870684 0.0497870684 5 0.006737947 0.8640390763 0.0067379439 0.006737947 9.9 5.017468206E-5 1207.799663 -0.1392914019 5.017463241E-5

I think you know where I’m going.

e^(-x^2) = 1 – x^2 + x^4/2! – x^6/3! + x^8/4! = Σ( (-x)^(2*n)/n!, from n = 0 to ∞)

 x = e^(-x^2) 10 terms 25 terms 50 terms 1 0.3678794412 0.3678794643 0.3678794412 0.3678794412 3 1.234098041E-4 442.2750223 -0.0118646275 1.234194001E-4 5 1.38879439E-11 18613495.8 -2834107793 85689.40174 9.9 2.72143414E-43 2.04347238E13 -3.10254183E24 7.951057508E34

(Something really goes bonkers as x increases and n increases)

Error Function
erf(x) = 2/√π * ∫(e^(-t^2) dt, 0, x)
= 2/√π * (x – x^3/3 + x^5/(5*2!) – x^7/(7*3!) + x^9/(9*4!) - ...)
= 2/√π * Σ( (-x^(2n+1)/((2n+1)*n!) from n = 0 to ∞ )

 x = erf(x) 10 terms 25 terms 50 terms 1 0.8427007929 0.8427007941 0.8427007929 0.8427007929 3 0.9999779095 68.58627744 0.9992050426 0.9999779095 5 1 4853382.901 -3070260210.4 4724.331354 9.9 1 1.076461715E13 -6.7395908E23 *overflows during calculation* (Result: 8.73442E33 from WolframAlpha) (erf(x) is practically 1 for x > 3)

Note: 9.9^(2*50+1) ≈ 3.623E100

Thoughts:

*  Taylor series are great when x is near its center point.  In the all the cases above, the center point is x = 0.

*  The more simple the expression, the better range of accuracy with less terms.

*  Before you recommend a Taylor Series to approximate f(x), check the accuracy and the range.  A cautionary tale.

Eddie

This blog is property of Edward Shore, 2016.

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